1. Section 3.5 – Transformation of Functions

2. Symmetry
Symmetric with respect to an axis: You can fold a graph along an axis and the graph will fall on top of itself. Each part of the graph is covered by its image.
Symmetric with respect to the origin: Rotating a graph 180⁰ about the origin results in the original graph. You can also fold along the x-axis AND the y-axis and the graph will fall on top of itself.

3. Symmetric with respect to 𝑥−axis
If the graph is symmetric with respect to the x-axis, the x-value stays the same!

4. Symmetric with respect to 𝑦−axis
If the graph is symmetric with respect to the y-axis, the y-value stays the same!

5. Symmetric With Respect to the Origin
If the graph is symmetric with respect to the origin, NOTHING stays the same!

6. Even and Odd Functions
Functions CAN’T be BOTH even and odd! They may be neither!

7. Even and Odd Functions
Determine whether the function is even, odd, or neither even nor odd.
NOT EVEN
NOT ODD
NEITHER

8. Even and Odd Functions
Determine whether the function is even, odd, or neither even nor odd.
NOT EVEN
ODD

9. Horizontal Translations
𝑓 𝑥 = 𝑥 2
𝑓 𝑥 = (𝑥+5) 2
𝑓 𝑥 = (𝑥 −3) 2
𝑓 𝑥 = 𝑥 2
𝑓 𝑥 = (𝑥+5) 2
𝑓 𝑥 = (𝑥 −3) 2

10. Horizontal Translations
𝑓 𝑥 = 𝑥 3
𝑓 𝑥 = (𝑥+2) 3
𝑓 𝑥 = (𝑥 −4) 3
𝑓 𝑥 = 𝑥 3
𝑓 𝑥 = (𝑥+2) 3
𝑓 𝑥 = (𝑥 −4) 3

11. Horizontal Translations
𝑓 𝑥 = 𝑥
𝒇 𝒙 = 𝒙−𝟏
𝒇 𝒙 = 𝒙+𝟑
𝒇 𝒙 = 𝒙
𝒇 𝒙 = 𝒙−𝟏
𝒇 𝒙 = 𝒙+𝟑

12. Horizontal Translations

13. VERTICAL Translations
𝑓 𝑥 = 3 𝑥
𝑓 𝑥 = 3 𝑥 +3
𝑓 𝑥 = 3 𝑥 −5
𝑓 𝑥 = 3 𝑥
𝑓 𝑥 = 3 𝑥 +3
𝑓 𝑥 = 3 𝑥 −5

14. VERTICAL Translations
𝑓 𝑥 = 1 𝑥
𝑓 𝑥 = 1 𝑥 −3
𝑓 𝑥 = 1 𝑥 +1
𝑓 𝑥 = 1 𝑥
𝑓 𝑥 = 1 𝑥 −3
𝑓 𝑥 = 1 𝑥 +1

15. VERTICAL Translations

16. REFLECTIONS ACROSS THE 𝑥−AXIS
𝑓 𝑥 = 1 𝑥 2
𝑓 𝑥 =− 1 𝑥 2
𝑓 𝑥 = 1 𝑥 2
𝑓 𝑥 =− 1 𝑥 2

17. REFLECTIONS ACROSS THE 𝑥−AXIS
𝑓 𝑥 = 3 𝑥
𝑓 𝑥 =− 3 𝑥
𝑓 𝑥 = 3 𝑥
𝑓 𝑥 =− 3 𝑥

18. REFLECTIONS ACROSS THE 𝑥−AXIS

19. VERTICAL STRETCHING OR SHRINKING
𝑓 𝑥 = 𝑥
𝑓 𝑥 =6 𝑥
𝑓 𝑥 = 1 3 𝑥
𝑓 𝑥 = 𝑥
𝑓 𝑥 =6 𝑥
𝑓 𝑥 = 1 3 𝑥

20. VERTICAL STRETCHING OR SHRINKING
𝑓 𝑥 = 𝑥
𝑓 𝑥 = 𝑥
𝑓 𝑥 =4 𝑥
𝑓 𝑥 = 𝑥
𝑓 𝑥 = 𝑥
𝑓 𝑥 =4 𝑥

21. VERTICAL STRETCHING OR SHRINKING

22. Transformations
Describe the transformations associated with the function and then graph the function.
𝑓 𝑥 = (𝑥+1) 2 −5
Basic Function: 𝑓 𝑥 = 𝑥 2
Shift 1 unit to the left
Shift down 5 units

23. Transformations
Describe the transformations associated with the function and then graph the function.
𝑓 𝑥 =− 3 𝑥−4 +1
Basic Function: 𝑓 𝑥 = 3 𝑥
Shift 4 units to the right
Reflect over 𝑥−axis
Shift up 1 unit

24. Transformations